Simulation of spatial systems with demographic noise

The demographic (shot) noise in population dynamics scales with the square root of the population size. This process is very important, as it yields an absorbing state at zero field, but simulating it, especially on spatial domains, is a nontrivial task. Here, we analyze two similar methods that were suggested for simulating the corresponding Langevin equation, one by Pechenik and Levine and the other by Dornic, Chaté, and Muñoz (DCM). These methods are based on operator-splitting techniques and the essential difference between them lies in which terms are bundled together in the splitting process. Both these methods are first order in the time step so one may expect that their performance will be similar. We find, surprisingly, that when simulating the stochastic Ginzburg-Landau equation with two deterministic metastable states, the DCM method exhibits two anomalous behaviors. First, the stochastic stall point moves away from its deterministic counterpart, the Maxwell point, when decreasing the noise. Second, the errors induced by the finite time step are larger by a significant factor (i.e., >10×) in the DCM method. We show that both these behaviors are the result of a finite-time-step induced shift in the deterministic Maxwell point in the DCM method, due to the particular operator splitting employed. In light of these results, care must be exercised when computing quantities like phase-transition boundaries (as opposed to universal quantities such as critical exponents) in such stochastic spatial systems.